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Early History of Metal Theory

Early History of Metal Theory 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …) The Basic Hamiltonian Approximations & Assumptions The Ground State ( T = 0) Wave-functions, allowed states, Fermi sphere, density of states Thermal Properties

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Early History of Metal Theory

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  1. Early History of Metal Theory 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …) The Basic Hamiltonian Approximations & Assumptions The Ground State (T = 0) Wave-functions, allowed states, Fermi sphere, density of states Thermal Properties Expectation values, energy, specific heat Electrical Transport Properties DC and AC conductivities Magnetic Properties Classical Hall effect, Pauli paramagnetism, Landau diamagnetism, cyclotron resonance, the quantum Hall effect Chapter 1: The Free Electron Fermi Gas

  2. Drude-Lorentz Model: 1900-1904 • Ashcroft-Mermin Chapter 1: • Based on the discovery of electrons by J. J. Thomson (1897) • Electrons as particles (Newton’s equation) • Electron “gas” (Maxwell-Boltzmann statistics) • Worked well • Explained Wiedemann-Franz law (1853) k/sT ~ 2-3 x 10-8 (W-ohm/K2) (by double mistakes) • Could not account for: • T-dependence of k alone • T-dependence of s alone • Electronic specific heat ce (too big) • Magnetic susceptibility cm (too big) Paul Drude (1863-1906) Hendrik Lorentz (1853-1928)

  3. Fermi-Dirac Statistics: 1926 m Enrico Fermi (1901-1954) Paul Dirac (1902-1984) Shown by Fermi and Dirac independently  Opened a way to a realistic theory of metals

  4. Pauli Susceptibility: 1927 First successful application of FD statistics to metal theory : Pauli Wolfgang Pauli (1900-1958) 26 meV at 300 K ~5 eV  Explains why the classical result is too big

  5. Sommerfeld Model: 1928 • Ashcroft-Mermin Chapter 2: • Systematically recast Drude-Lorentz theory in terms of FD statistics rather than MB statistics • Wiedemann-Franz law (still) came out right • Estimated specific heat right • Difficulties remained: • Sign of Hall coefficient • Magneto-resistance • What determines the scattering time t? • What determine the density n? • Why are some elements non-metals? Arnold Sommerfeld (1868-1951)

  6. Bloch Theory: 1928 • Ashcroft-Mermin Chapters 8-10: • A major breakthrough in solid state theory • Took into account lattice periodic potential • Still treated electrons independently • Major accomplishments: • Meaning of t: lattice imperfections (phonons, defects, impurities, dislocations) • Concept of energy “bands” • Distinction between metals and insulators (also by A. H. Wilson in 1931) • Meaning of “holes”  positive Hall coefficient Felix Bloch (1905-1983)

  7. The Birth of ‘Fermiology’: 1930 • Ashcroft-Mermin Chapter 14: • Landau – Landau levels (1930)  predicted oscillations in c vs. H  period has info on the shape of the Fermi surface • Observations of Shubnikov-de Haas oscillations & de Haas-van Alphen oscillations (~1930) • Cyclotron resonance • Predicted by Dingle (1951) • First observed by Dresselhaus et al. (1953) • Theory refined by Luttinger & Kohn (1955,56) Lev Landau (1908-1968)

  8. Early History of Metal Theory 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …) The Basic Hamiltonian Approximations & Assumptions The Ground State (T = 0) Wave-functions, allowed states, Fermi sphere, density of states Thermal Properties Expectation values, energy, specific heat Electrical Transport Properties DC and AC conductivities Magnetic Properties Classical Hall effect, Pauli paramagnetism, Landau diamagnetism, cyclotron resonance, the quantum Hall effect Chapter 1: The Free Electron Fermi Gas

  9. The Basic Hamiltonian … kinetic energies … potential energies N nuclei (positive ions): Total charge = +NZae e- e- e- e- NZa electrons: Total charge = -NZae e- e- e- e- e- e- e- Za: atomic number N ~ 1023

  10. The Schrödinger Equation N ~ 1023 Massive many-body problem Exact solutions cannot be expected

  11. Early History of Metal Theory 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …) The Basic Hamiltonian Approximations & Assumptions The Ground State (T = 0) Wave-functions, allowed states, Fermi sphere, density of states Thermal Properties Expectation values, energy, specific heat Electrical Transport Properties DC and AC conductivities Magnetic Properties Classical Hall effect, Pauli paramagnetism, Landau diamagnetism, cyclotron resonance, the quantum Hall effect Chapter 1: The Free Electron Fermi Gas

  12. Assume M : constant The Static Lattice Approximation Constant potential  0

  13. Consider only free electrons The Free Electron Approximation Zvalence electrons per nucleus  free (or conduction) electrons (Za – Z) tightly-bound electrons per nucleus

  14. The Uniform-Background Approximation e- e- e- e- e- e- e- e- e- e- e- • Uniform distribution of positive ions • Neglect the discreteness of the ions But, constant potential  0 e- e- e- e- e- e- e- e- e- e- e- : ion

  15. The Independent Electron Approximation But, Requirement 1: Electrons are still confined to a volume V  allowed states Requirement 2: Electrons obey FD statistics  ground state construction

  16. The Simplified Hamiltonian in the absence of an external field Separable into NZ terms neglect i solve

  17. Early History of Metal Theory 1900-1930 (Drude, Lorentz, Fermi, Dirac, Pauli, Sommerfeld, Bloch, …) The Basic Hamiltonian Approximations & Assumptions The Ground State (T = 0) Wave-functions, allowed states, Fermi sphere, density of states Thermal Properties Expectation values, energy, specific heat Electrical Transport Properties DC and AC conductivities Magnetic Properties Classical Hall effect, Pauli paramagnetism, Landau diamagnetism, cyclotron resonance, the quantum Hall effect Chapter 1: The Free Electron Fermi Gas

  18. Ground-State (T = 0) Properties of Non-interacting Electrons Single-electron Schrödinger equation: We consider the metal to be a cube of side The electron is confined in this cube. • Procedure: • Find the energy levels of a single electron • Fill these levels up in a manner consistent with the Pauli principle

  19. Born-Von Karman’s Periodic Boundary Conditions Idea: If the metal is sufficiently large, we should expect its bulk properties to be unaffected by the detailed configuration of its surface.

  20. Solution B.C.

  21. Number of Allowed States The volume of k-space per point: A region of k-space of volume W will contain allowed values of k. The number of allowed k-values per unit volume of k-space (a.k.a. the k-space density of levels) spin degeneracy

  22. Construction of the Ground State: the Fermi Sphere n = N/V

  23. Fermi Surface in a Real Metal “Fermiology”: The study of the shape of Fermi surfaces in metals

  24. Fermi Wave Vector, Momentum, Energy, Velocity, and Temperature n ~ 1022 cm-3 ~1 Å-1 The Fermi momentum: ~10-23 kg-m/s ~5 eV The Fermi velocity: ~106 m/s ~104 K The Fermi wave vector: The Fermi energy: The Fermi temperature:

  25. Density of States (DOS) To calculate thermodynamic quantities, one needs to carry out summations of the type over allowed values, i.e., For mathematical convenience, want to convert sums into integrals k-space DOS: Dk

  26. Density of States (DOS) In many cases, k-dependence appears only through Namely, So, want to write D(): Energy DOS – the number of allowed states having energies between  and  + d Note: is the DOS per unit volume

  27. Density of States (DOS) Want to calculate D(). Easier in spherical coordinates: So,

  28. Density of States (DOS) Exercise: derive the exact form of DOS for each case

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